Re: Linear Programming Assignment Guidance? (Email/PM)

Here is the word problem.

You are the manager at Nanuq Australia’s Hill Valley factory, which produces edge trimmers, leaf blowers and mowers.
This morning, your boss, the regional manager, met with you and reluctantly explained that the company is struggling, and some contraction will be necessary.
In particular, you have been informed that, unless your company can make a profit of at least $250 000 in the next 29 working days, it will be closed down.
Resource requirements and profits per unit of production are as follows.

Edge trimmer

Leaf blower

Mower

Labour (minutes)

4

6

9

Energy (kWh)

1.5

2

5

Plastic (kg)

1.5

2

0.5

Copper (kg)

0.4

0.8

2.4

Steel (kg)

1.2

0.3

30

Profit per unit

$18

$24

$101

Table 1: Resource requirements and profits per unit of production.

Over the planning period (that is, the next 29 working days),
there will be 550 hours of labour, 15MWh of energy, 6.24 tonnes of copper and 63.06 tonnes of steel available at the Riverton factory.

The amount of plastic available for the planning period is not known exactly.
Your factory buys plastic in an unfinished form and uses a production line to prepare the plastic components needed in production of the edge trimmers, leaf blowers and mowers.

Unfortunately, the equipment in the plastic production line is old and tends to break down, reducing the total amount of finished plastic available.

In particular, each day, the plastic production line is either functioning or broken down.

If it is broken down on one day, then it will be fixed and functioning again the following day, and the required repairs will cost the factory $200 per breakdown.
On the other hand, if it is functioning one day, then on the following day it will either continue functioning or break down, with probabilities that depend on how recently it was fixed:

Probability that today the production line will be:

Days since last breakdown:

Functioning:

Broken down:

2–5:

0.875

0.125

6 or more:

0.75

0.25

Table 2: Probabilities of the plastic production line breaking down today if it was functioning
yesterday.

You can assume that the plastic production line was fixed the day before the planning period starts, so that it is definitely functioning on the first day of the planning period.

It produces 200 kg of finished plastic per functioning day, so that the maximum amount available for production of edge trimmers,
leaf blowers and mowers is 5.8 tonnes (if no breakdowns occur).

Work through the following problems to estimate the probability that the factory will be shut down.

For Problem 2, the numerical answer is given (to allow you to approach Problem 3 with confidence), but you must show how to derive that numerical answer.

Re: Linear Programming Assignment Guidance? (Email/PM)

The first two questions.

Problem 1
Formulate a linear program to maximise profit during the 29-day planning period, under the assumption that no breakdowns occur.

Problem 2
Enter the linear program into Excel, and use the Solver to determine the optimal production plan and
the resulting profit. Also, demonstrate that the shadow price of finished plastic is $7.09 per kilogram.

I think I managed to formulate the linear program without difficulty.

Maximise P = 18Xet + 24Xlb + 101Xm

Xet, number of edge trimmers produced
Xlb, number of leaf blowers produced
Xm, number of mowers produced

Re: Linear Programming Assignment Guidance? (Email/PM)

I started having problems when my sensitivity analysis showed a finished plastic shadow price of 11(something).
I understand shadow prices are the extra profit/loss per unit increase.
But I was lost as to where 7.09 came from?